## 海峡两岸算术几何研讨会

Cross Strait Workshop on Arithmetic Geometry

Wuhan University

Dec. 16-18, 2016

### Main Organizers

 Chia-Fu Yu Jiangwei Xue

### Invited Speakers

 Lei Fu Yau Mathematical Sciences Center Ming-Hsuan Kang National Chiao Tung University Wen-Ching Winnie Li Pennsylvania State University Ruochuan Liu Beijing International Center for Math Research Hourong Qin Nanjing University Mao Sheng University of Science and Technology of China Ye Tian Chinese Academy of Sciences Chian-Jen Wang Tamkang University Fei Xu Capital Normal University Jiangwei Xue Wuhan University Yi-Fan Yang National Chiao Tung University Chia-Fu Yu Academia Sinica

### Information

Program.pdf
Campus Map

Registration: 10: 00-20:00, Dec. 15, 2016
Accommodation: All nonlocal guests are accommodated on campus in Luojia Shanzhuang (珞珈山庄, Tel: 86-27-6875 2935). Please see the campus map above.
Lecture Room: All talks will be held in the lecture room on the 3rd floor of the School of Mathematics and Statistics building.

## Program

Time: Dec.16, 08:50-09:40

### From K3 surfaces to noncongruence modular forms

#### Wen-Ching Winnie Li

Abstract: In this talk we shall tell the story how the modularity of K3 surfaces led to the modularity or potential automorphy of an infinite family of Scholl representations attached to noncongruence modular forms. This is the first family of Scholl representations with unbounded degree for which the potential automorphy is established. The talk is based on a joint work with Tong Liu and Ling Long.

Time: Dec.16, 09:50-10:40

### Arithmetic purity of strong approximation

#### Fei Xu

Abstract: It is well-known that weak approximation is birational invariant between smooth varieties over number field by implicit function theorem. This is not true for strong approximation. For example, if a smooth variety satisfies strong approximation, then this variety must be simply connected. As Nagata-Zariski purity theorem, one can expect that if a smooth variety satisfies strong approximation, so does any open sub-variety with the codimension of the complement greater than 1. In this talk, we explain that this is true for semi-simple simply connected quasi-split linear algebraic groups. This is part of our joint program with Yang Cao and Yongqi Liang.

Time: Dec.16, 11:10-12:00

### The Lang-Trotter conjecture for CM elliptic curves

#### Hourong Qin

Abstract: Let $E$ be an elliptic curve over $\mathbb{Q}.$ For a fixed integer $r,$ define the prime-counting function $\pi_{E,r}(x):=\sum_{p\leq x, a_E,a_p=r}1$. If $r=0$, then assume additionally that $E$ has no complex multiplication. The Lang-Trotter conjecture predicts that $$\pi_{E,r}(x)=C_{E,r}\cdot \frac{\sqrt{x}}{{\rm log}x}+o\left(\frac{\sqrt{x}}{{\rm log}x}\right)$$ as $x\longrightarrow \infty,$ where $C_{E,r}$ is a specific non-negative constant.

It is open whether there exists a polynomial in one variable of degree $2$ that represents infinitely many primes. For example, at present, we do not know whether the polynomial $x^2+1$ represents infinitely many primes. The Hardy-Littlewood conjecture gives a similar asymptotic formula as above for the number of primes of the form $ax^2+bx+c$.

We establish a relationship between the Hardy-Littlewood conjecture and the Lang-Trotter conjecture for CM elliptic curves.

Time: Dec.16, 14:30-15:20

### Rational torsion points on the generalised Jacobian of a modular curve with cuspidal modulus

#### Yi-Fan Yang

Abstract: In this talk we will consider the generalised Jacobian of the modular curve $X_0(N)$ with respect to the reduced divisor given by the sum of all cusps. When $N$ is a prime power greater than 3, we find that the group of rational torsion points on the generalised Jacobian tends to be much smaller than that of the classical Jacobian. This is a joint work with Takao Yamazaki.

Time: Dec.16, 15:35-16:25

### Finiteness of cohomology of local systems on rigid analytic spaces

#### Ruochuan Liu

Abstract: In a recent joint work with Kedlaya, we prove that the cohomology groups of an étale $\mathbb{Q}_p$-local system on a smooth proper rigid analytic space are finite-dimensional $\mathbb{Q}_p$-vector spaces, provided that the base field is either a finite extension of $\mathbb{Q}_p$ or an algebraically closed nonarchimedean field containing $\mathbb{Q}_p$. This result manifests as a special case of a more general finiteness result for the higher direct images of a relative $(\phi, \Gamma)$-module along a smooth proper morphism of rigid analytic spaces over a mixed-characterstic nonarchimedean field.

Time: Dec.16, 16:40-17:30

### CM-fields as endomorphism algebras of abelian varieties over finite fields

#### Jiangwei Xue

Abstract: Given a CM-field $K$, one may ask whether there exists an abelian variety $A$ over some finite field $\mathbb{F}_q$ with $\mathrm{End}^0(A)=K$. We show the answer is affirmative for most CM-fields except two special cases, and there exist infinitely many CM-fields that are not realizable as endomorphism algebras over any finite field. This is a joint work with Chia-Fu Yu.

Time: Dec.17, 09:00-09:50

### GOLDFELD CONJECTURE FOR CONGRUENT ELLIPTIC CURVES

#### Ye Tian

Abstract: A positive integer $n$ is called a congruent number if it is the area of a right triangle with rational side lengths, or equivalently the elliptic curve $E_n: ny^2=x^3-x$ has Mordell-Weil group of rank $>0$. Note that for square-free positive integer $n$, the L-function $L(E_n, s)$ has sign $-1$ if and only if $n\equiv 5, 6,7 \mod{8}$. Goldfeld conjectured that among all positive square-free integers congruent to 5, 6, 7 modulo 8, those $n$ with $\mathrm{ord}_{s=1}L(E_n, s)=1$ has density one. In this talk, we prove that this density is $>50\%$. This talk is based on our joint work with Xinyi Yuan and Shouwu Zhang, and work of Smith.

Time: Dec.17, 10:05-10:55

### On a problem of Illusie in Hodge theory over positive characteristic

#### Mao Sheng

Abstract: In this talk, I will report our recent study towards an open problem posed by Professor Luc Illusie. This problem is about the analogue of the fundamental decomposability theorem of Deligne-Illusie in the semi-stable reduction case. It turns out that it has negative answers in relative dimensions greater than or equal to two and in all positive characteristics. This is based on a joint work with Junchao Shentu.

Time: Dec.17, 11:10-12:00

### Alternating product of twisted Poincare series

#### Ming-Hsuan Kang

Abstract: Poincare series is a generating function of an affine Weyl group. It is a rational function and encodes the degrees of polynomial invariants. One can also define the Poincare series on every parabolic subgroup. In this talk, we consider the alternating product of the Poincare series of all parabolic subgroup with respect to their ranks. We will show that the reciprocal of the alternating product is indeed a polynomial of certain special form. Moreover, for the case of affine rank two, we find a geometric interpretation of the alternating product as the gallery zeta function on some complex.

Time: Dec.17, 14:30-15:20

### Algebraic geometry on loop spaces

#### Lei Fu

Abstract: I will introduce some geometric concepts on loop spaces and try to do arithmetic geometry on models of them.

Time: Dec.17, 15:35-16:25

### Zeta functions of finite complexes arising from groups of rank two

#### Chian-Jen Wang

Abstract: Let $F$ be a non-archimedean local field and let $G$ be a reductive group of rank two defined over $F$. This talk will survey recent progress toward introducing and understanding zeta functions of finite complexes arising from the Bruhat-Tits buildings of $G$. We shall discuss the combinatorial interpretation of these zeta functions and their connections to local Langlands L-functions. This is based on joint work with Ming-Hsuan Kang and Winnie Li.

Time: Dec.17, 16:40-17:30

### On superspecial abelian surfaces and type numbers

#### Chia-Fu Yu

Abstract: Classical results of Deuring and Eicher gave explicit formulas for the class and type numbers of the definite rational quaternion algebra ramified at p and infinity. These numbers correspond to the numbers of isomorphism classes of supersingular elliptic curves in characteristic p and their endomorphism algebras. In this talk i will explain joint work with Jiangwei Xue, which generalizes these classical results from supersingular elliptic curves to superspecial abelian surfaces. We plan also to address certain connection with arithmetic genera of Hilbert modular surfaces.

## Winter School

### Winter School on Shimura varieties and related topics in Wuhan

Dec.9 (Fri.)- Dec.12 (Mon.)

Before the Workshop, Prof. Chia-Fu Yu and I are also organizing a winter school on Shimura varieties and related topic in Wuhan. We are honored to have Ke Chen (Nanjing University), Xu Shen(Academy of Mathematics and Systems Science), Chia-Fu Yu (Academia Sinica) and Chao Zhang (Tsinghua University) as our speakers. The initial framework of the Winter school is as follows:

1. Ke Chen, Special subvarieties and André-Oort conjecture (2 lectures)
2. Xu Shen, Uniformization of Shimura varieties (4 lectures)
3. Chia-Fu Yu, Deligne's absolute Hodge cycles (3 lectures)
4. Chao Zhang, EO stratification (3 lectures)

All topics are tentative, may be changed or modified as the date approaches.